2018-02-22T08:55:52Zhttp://toc.ui.ac.ir/?_action=export&rf=summon&issue=38762017-03-0110.22108Transactions on CombinatoricsTrans. Comb.2251-86572251-8657201761On annihilator graph of a finite commutative ringSanghitaDuttaChanlemkiLanong‎The annihilator graph $AG(R)$ of a commutative ring $R$ is a simple undirected graph with the vertex set $Z(R)^*$ and two distinct vertices are adjacent if and only if $ann(x) cup ann(y)$ $ neq $ $ann(xy)$‎. ‎In this paper we give the sufficient condition for a graph $AG(R)$ to be complete‎. ‎We characterize rings for which $AG(R)$ is a regular graph‎, ‎we show that $gamma (AG(R))in {1,2}$ and we also characterize the rings for which $AG(R)$ has a cut vertex‎. ‎Finally we find the clique number of a finite reduced ring and characterize the rings for which $AG(R)$ is a planar graph‎.‎Annihilator‎‎Clique number‎‎Domination Number20170301111http://toc.ui.ac.ir/article_20360_56c78d48b767dab5eff9143a4cf11336.pdf2017-03-0110.22108Transactions on CombinatoricsTrans. Comb.2251-86572251-8657201761A neighborhood union condition for fractional $(k,n',m)$-critical deleted graphsYunGaoMohammad RezaFarahaniWeiGaoA graph $G$ is called a fractional‎ ‎$(k,n',m)$-critical deleted graph if any $n'$ vertices are removed‎ ‎from $G$ the resulting graph is a fractional $(k,m)$-deleted‎ ‎graph‎. ‎In this paper‎, ‎we prove that for integers $kge 2$‎, ‎$n',mge0$‎, ‎$nge8k+n'+4m-7$‎, ‎and $delta(G)ge k+n'+m$‎, ‎if‎ ‎$$|N_{G}(x)cup N_{G}(y)|gefrac{n+n'}{2}$$‎ ‎for each pair of non-adjacent vertices $x$‎, ‎$y$ of $G$‎, ‎then $G$‎ ‎is a fractional $(k,n',m)$-critical deleted graph‎. ‎The bounds for‎ ‎neighborhood union condition‎, ‎the order $n$ and the minimum degree‎ ‎$delta(G)$ of $G$ are all sharp‎.‎Graph‎‎fractional‎ ‎factor‎‎fractional $(kn&#039;m)$-critical deleted graph‎‎neighborhood‎ ‎union condition201703011319http://toc.ui.ac.ir/article_20355_2293d2e8b5527d56f39b0d5e01456cad.pdf2017-03-0110.22108Transactions on CombinatoricsTrans. Comb.2251-86572251-8657201761The condition for a sequence to be potentially $A_{L‎, ‎M}$‎- graphicShariefuddinPirzadaBilalA. ChatThe set of all non-increasing non-negative integer sequences $pi=(d_1‎, ‎d_2,ldots,d_n)$ is denoted by $NS_n$‎. ‎A sequence $piin NS_{n}$ is said to be graphic if it is the degree sequence of a simple graph $G$ on $n$ vertices‎, ‎and such a graph $G$ is called a realization of $pi$‎. ‎The set of all graphic sequences in $NS_{n}$ is denoted by $GS_{n}$‎. ‎The complete product split graph on $L‎ + ‎M$ vertices is denoted by $overline{S}_{L‎, ‎M}=K_{L} vee overline{K}_{M}$‎, ‎where $K_{L}$ and $K_{M}$ are complete graphs respectively on $L = sumlimits_{i = 1}^{p}r_{i}$ and $M = sumlimits_{i = 1}^{p}s_{i}$ vertices with $r_{i}$ and $s_{i}$ being integers‎. ‎Another split graph is denoted by $S_{L‎, ‎M} = overline{S}_{r_{1}‎, ‎s_{1}} veeoverline{S}_{r_{2}‎, ‎s_{2}} vee cdots vee overline{S}_{r_{p}‎, ‎s_{p}}= (K_{r_{1}} vee overline{K}_{s_{1}})vee (K_{r_{2}} vee overline{K}_{s_{2}})vee cdots vee (K_{r_{p}} vee overline{K}_{s_{p}})$‎. ‎A sequence $pi=(d_{1}‎, ‎d_{2},ldots,d_{n})$ is said to be potentially $S_{L‎, ‎M}$-graphic (respectively $overline{S}_{L‎, ‎M}$)-graphic if there is a realization $G$ of $pi$ containing $S_{L‎, ‎M}$ (respectively $overline{S}_{L‎, ‎M}$) as a subgraph‎. ‎If $pi$ has a realization $G$ containing $S_{L‎, ‎M}$ on those vertices having degrees $d_{1}‎, ‎d_{2},ldots,d_{L+M}$‎, ‎then $pi$ is potentially $A_{L‎, ‎M}$-graphic‎. ‎A non-increasing sequence of non-negative integers $pi = (d_{1}‎, ‎d_{2},ldots,d_{n})$ is potentially $A_{L‎, ‎M}$-graphic if and only if it is potentially $S_{L‎, ‎M}$-graphic‎. ‎In this paper‎, ‎we obtain the sufficient condition for a graphic sequence to be potentially $A_{L‎, ‎M}$-graphic and this result is a generalization of that given by J‎. ‎H‎. ‎Yin on split graphs‎.‎Split graph‎‎complete product split graph‎‎potentially $H$-graphic Sequences201703012127http://toc.ui.ac.ir/article_20361_5539a345ae0f45bb6974e8e9397a9145.pdf2017-03-0110.22108Transactions on CombinatoricsTrans. Comb.2251-86572251-8657201761Some properties of comaximal ideal graph of a commutative ringMehrdadAzadiZeinabJafariLet $R$ be a commutative ring with identity‎. ‎We use‎ ‎$varphi (R)$ to denote the comaximal ideal graph‎. ‎The vertices‎ ‎of $varphi (R)$ are proper ideals of R which are not contained‎ ‎in the Jacobson radical of $R$‎, ‎and two vertices $I$ and $J$ are‎ ‎adjacent if and only if $I‎ + ‎J = R$‎. ‎In this paper we show some‎ ‎properties of this graph together with planarity of line graph‎ ‎associated to $varphi (R)$‎.‎‎Comaximal graph‎‎planar graph‎‎line‎ ‎graph201703012937http://toc.ui.ac.ir/article_20429_cb19821e16c613c386c6392dde7a5d30.pdf2017-03-0110.22108Transactions on CombinatoricsTrans. Comb.2251-86572251-8657201761A family of $t$-regular ‎self-complementary $k$-hypergraphsMasoudAriannejadMojganEmamiOzraNaserianWe use the recursive method of construction large sets of t-designs given by Qiu-rong Wu (A note on extending t-designs‎, ‎{em Australas‎. ‎J‎. ‎Combin.}‎, ‎{bf 4} (1991) 229--235.), and present a similar method for constructing $t$-subset-regular‎ ‎self-complementary $k$-uniform hypergraphs of order $v$‎. ‎As an‎ ‎application we show the existence of a new family of 2-subset-regular‎ ‎self-complementary 4-uniform hypergraphs with $v=16m+3$‎.Self-complementary‎ ‎hypergraph‎‎Uniform hypergraph‎‎Regular hypergraph‎‎Large sets of t-designs201703013946http://toc.ui.ac.ir/article_20363_caa3ab087951b3985516a80dc389ee3a.pdf2017-03-0110.22108Transactions on CombinatoricsTrans. Comb.2251-86572251-8657201761On the skew spectral moments of graphsFatemehTaghvaeeGholam HosseinFath-TabarLet $G$ be a simple graph‎, ‎and $G^{sigma}$‎ ‎be an oriented graph of $G$ with the orientation ‎$sigma$ and skew-adjacency matrix $S(G^{sigma})$‎. ‎The $k-$th skew spectral‎ ‎moment of $G^{sigma}$‎, ‎denoted by‎ ‎$T_k(G^{sigma})$‎, ‎is defined as $sum_{i=1}^{n}( ‎‎‎lambda_{i})^{k}$‎, ‎where $lambda_{1}‎, ‎lambda_{2},cdots‎, ‎lambda_{n}$ are the eigenvalues of $G^{sigma}$‎. ‎Suppose‎ ‎$G^{sigma_1}_{1}$ and $G^{sigma_2}_{2}$ are two digraphs‎. ‎If there‎ ‎exists an integer $k$‎, ‎$1 leq k leq n-1$‎, ‎such that for each‎ ‎$i$‎, ‎$0 leq i leq k-1$‎, ‎$T_i(G^{sigma_1}_{1}) =‎ ‎T_i(G^{sigma_2}_{2})$ and‎ ‎$T_k(G^{sigma_1}_{1}) ‎‎Oriented graph‎‎skew spectral moment‎‎skew eigenvalue‎‎$T$-order‎‎skew characteristic polynomial201703014754http://toc.ui.ac.ir/article_20737_9c81a151b424aac06fc6253943dc89a2.pdf